Jennifer Lopez and Alex Rodriguez say reports of their demise as a couple have been greatly exaggerated — well, maybe just exaggerated — because they say they are still fighting to stay together.

Both JLo and A-Rod, “All the reports are inaccurate. We are working through some things.” We’re also told what they’re working through has nothing to do with a third party.

Sources connected to the couple made it clear to TMZ and other outlets Friday … the couple had called off their engagement and broken up. We asked our sources with direct knowledge why we were all told they broke up, and we were told things were “bad” on Friday.

As of now, A-Rod’s still in Miami and JLo’s in the Dominican Republic shooting a movie, and it’s unclear when they’ll be back together. FYI, it’s a hop, skip and a jump from Miami to the D.R.

There’s a lot at stake in their relationship. Aside from them, they both have kids that have effectively blended as a big family, so that’s got to come into play as they navigate their relationship.

They’ve been together for 4 years, and for most of the time they seemed tight. To be continued …